Synthetic Time Series Generation via Complex Networks

The gist Synthetic time series generation via complex networks investigates using an inverse mapping framework called Inverse Quantile Graph (InvQG) to generate synthetic time series, showing it effectively preserves marginal distributions and short-term temporal dependencies while exhibiting predictable limitations in capturing long-range or higher-order dynamics

Introduction and Motivation

Time series data are essential for a wide range of applications, yet access to high-quality datasets is often constrained by privacy concerns, acquisition costs, and labelling challenges Synthetic time series generation has emerged as a promising approach to address these limitations A central challenge, however, is to generate synthetic data that simultaneously preserve essential statistical and temporal properties of the original series while remaining useful for downstream analytical tasks such as clustering, classification or forecasting Deep generative models like Generative Adversarial Networks (GANs) have become a dominant approach but typically require extensive training and may suffer from instability and limited interpretability This work investigates the use of complex network mappings for synthetic time series generation, focusing on the Quantile Graph (QG) representation and its inverse

Methodology: Inverse Quantile Graph (InvQG)

The proposed method introduces the Inverse Quantile Graph (InvQG) framework as a practical and interpretable method for generating synthetic time series based on transition dynamics encoded in quantile space The approach involves a 2-step process: 1) Network-Domain Mapping, which uses transition probabilities to transform a time series Y into a graph G = (V, E,W) and 2) Time-Series Reverse Mapping (Reconstruction), which defines an inverse mapping M−1: G −→ T to reconstruct a synthetic time series Y∗ from the Markov transition matrix W representing graph G

The InvQG mapping involves the following steps

1. Extract the adjacency (Markov transition) matrixW of the quantile graph produced by QG mapping

2. Retrieve the sample quantiles of the original time series

3. Select an initial quantile qk at random among those with non-zero outgoing probability

4. For t = 1,..., T: Determine the value range of the current quantile qk and sample a value uniformly from that range

5. Transition to the next quantile qk′ by sampling according to the non-zero probabilities in row k of W

Empirical Evaluation and Fidelity Analysis

The empirical evaluation follows a structured and paired design to assess both fidelity and utility of synthetic time series generated by InvQG, as well as to benchmark its performance against alternative generative approaches The evaluation comprises four complementary components: (i) statistical feature preservation, (ii) network-based topological similarity, and (iii) utility in downstream clustering and classification tasks

Statistical Fidelity Analysis shows that across most models and features, the distributions of the paired differences are centered around zero, indicating that the InvQG preserves the principal statistical characteristics of the original series This is particularly evident for trend-related measures, entropy, and short-term autocorrelation (lag 1), demonstrating that the method effectively captures marginal distributions and immediate temporal dependencies across a broad range of time series models Systematic deviations emerge primarily for models characterized by strong periodicity or long-range dependence, notably AR2, ARFIMA, ARIMA, and HMM processes The underlying cause is structural: InvQG relies on a first-order Markov transition model and therefore encodes dependencies only between consecutive observations

Utility Analysis and Benchmarking

Utility via Network Representations is evaluated using topological features derived from time-series networks computed with the NetF framework, as well as clustering and classification performance based on these features Topological Feature Consistency shows that for most models, InvQG preserves the structural signatures captured by network representations, which are essential for distinguishing between different types of temporal behavior Clustering-Based Utility indicates that clustering performance on the combined original-synthetic dataset remains strong and close to that obtained using original data alone Furthermore, in a Train on Synthetic, Test on Real (TSTR) paradigm, KNN achieves near-perfect performance (∆Acc = −0.002), while RF and SVM show moderate degradation

Benchmarking against Generative Models shows that InvQG produces synthetic data whose empirical distributions closely overlap with those of the original series across most models In contrast, TimeGAN and DoppelGANger display less consistent behavior, with TimeGAN frequently generating synthetic samples that form clusters partially separated from the original data

Computational Performance and Robustness

The computational complexity of the InvQG method is O(QT + Q 2), which simplifies to O(QT) in practice, making it more computationally efficient than GAN-based approaches like TimeGAN and DoppelGANger, which require iterative training procedures The method is robust to the choice of Q within a wide and practically relevant range—in particular, the literature suggests the rule Q = 2T1/3 Moderate values of Q provide a good trade-off between fidelity and computational efficiency, confirming that the method is stable

Overall, InvQG positions itself as a competitive alternative for synthetic time series generation by offering a transparent mechanism that balances fidelity, interpretability, and efficiency Future work may explore extensions of the framework to address its current limitations, including the incorporation of higher-order transition structures

Conclusion

InvQG achieves high statistical fidelity across a diverse set of time series models by effectively preserving marginal distributions and short-term temporal dependencies When compared with state-of-the-art GAN-based approaches, InvQG shows competitive performance in reproducing the empirical distribution of the data while offering substantial advantages in terms of interpretability, computational efficiency, and ease of use The method is robust to the choice of Q within a wide and practically relevant range, confirming its practical applicability However, due to its reliance on a first-order transition framework, InvQG has a limited ability to capture long-range or higher-order temporal dependencies The results position InvQG as a competitive, efficient, and interpretable alternative for synthetic time series generation

Available at https://github.

1. Bold header: High-Fidelity, Interpretability Synthetic Data Generation

InvQG provides a simple yet effective framework for synthetic time series generation that effectively preserves marginal distributions and short-term temporal dependencies across a broad range of time series models, making it a transparent alternative to deep generative models.

2. Bold header: Robustness to Model Complexity

The method is shown to be robust across various statistical families, as the analysis confirms that InvQG preserves the main statistical properties of the original data across a broad range of quantile resolutions and is robust to the choice of Q within a wide and practically relevant range.

3. Bold header: Efficient Real-Time Data Augmentation

The computational complexity is favorable, with InvQG having an O(QT + Q 2) complexity, which simplifies to O(QT) in practice, offering a solution where fast data generation is required compared to GANs that require iterative training.

4. Bold header: Enhanced Clustering and Classification Utility

Synthetic series are shown to remain useful for downstream tasks, as the network-based analysis indicates that clustering performance on the combined original-synthetic dataset remains strong and close to [the performance obtained using original data alone].

5. Bold header: Benchmarking Against State-of-the-Art

InvQG achieves competitive distributional fidelity relative to the considered GAN-based approaches on controlled synthetic data, while offering substantial advantages in terms of interpretability, computational efficiency, and ease of use.